#8095578

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2,5,6)(3,4,7,8)(13,21)(18,22)(19,24)(20,23)(27,33)(30,34)(35,39)(38,40);(2,6)(3,7)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26,29,32)(27,34,33,30)(28,36,37,31)(35,38,39,40);(1,3)(2,4)(5,7)(6,8)(10,15)(12,16)(18,22)(20,23)(25,27,29,33)(26,30,32,34)(28,35,37,39)(31,38,36,40);(1,4)(2,3)(5,8)(6,7)(9,11,14,17)(10,16,15,12)(13,19,21,24)(18,23,22,20)(25,28,29,37)(26,31,32,36)(27,35,33,39)(30,38,34,40);(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002774);